Graph partitions and integer flows

We present improved upper bounds on the spanning ratio of constrained [Formula: see text]-graphs with at least 6 cones and constrained Yao-graphs with 5 or at least 7 cones. Given a set of points in the plane, a Yao-graph partitions the plane around each vertex into [Formula: see text] disjoint cones, each having aperture [Formula: see text], and adds an edge to the closest vertex in each cone. Constrained Yao-graphs have the additional property that no edge properly intersects any of the given line segment constraints. Constrained [Formula: see text]-graphs are similar to constrained Yao-graphs, but use a different method to determine the closest vertex. We present tight bounds on the spanning ratio of a large family of constrained [Formula: see text]-graphs. We show that constrained [Formula: see text]-graphs with [Formula: see text] ([Formula: see text] and integer) cones have a tight spanning ratio of [Formula: see text], where [Formula: see text] is [Formula: see text]. We also present improved upper bounds on the spanning ratio of the other families of constrained [Formula: see text]-graphs. These bounds match the current upper bounds in the unconstrained setting. We also show that constrained Yao-graphs with an even number of cones ([Formula: see text]) have spanning ratio at most [Formula: see text] and constrained Yao-graphs with an odd number of cones ([Formula: see text]) have spanning ratio at most [Formula: see text]. As is the case with constrained [Formula: see text]-graphs, these bounds match the current upper bounds in the unconstrained setting, which implies that like in the unconstrained setting using more cones can make the spanning ratio worse.

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Destroying Noncomplete Regular Components in Graph Partitions

Journal of Graph Theory ◽

10.1002/jgt.21634 ◽

2012 ◽

Vol 72 (2) ◽

pp. 123-127 ◽

Cited By ~ 1

Author(s):

Landon Rabern

Keyword(s):

Graph Partitions

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Integer flows

Journal of Graph Theory ◽

10.1002/jgt.3190070307 ◽

1983 ◽

Vol 7 (3) ◽

pp. 349-357 ◽

Cited By ~ 32

Author(s):

D. H. Younger

Keyword(s):

Integer Flows

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FGPGA: An efficient genetic approach for producing feasible graph partitions

2015 International Conference on Networking Systems and Security (NSysS) ◽

10.1109/nsyss.2015.7043513 ◽

2015 ◽

Cited By ~ 1

Author(s):

Md. Lisul Islam ◽

Novia Nurain ◽

Swakkhar Shatabda ◽

M Sohel Rahman

Keyword(s):

Genetic Approach ◽

Graph Partitions

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The Price of Order

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195916600013 ◽

2016 ◽

Vol 26 (03n04) ◽

pp. 135-149

Author(s):

Prosenjit Bose ◽

Pat Morin ◽

André van Renssen

Keyword(s):

Lower Bounds ◽

Large Family ◽

Worst Case ◽

Graph Partitions ◽

First Results

We present tight bounds on the spanning ratio of a large family of ordered [Formula: see text]-graphs. A [Formula: see text]-graph partitions the plane around each vertex into [Formula: see text] disjoint cones, each having aperture [Formula: see text]. An ordered [Formula: see text]-graph is constructed by inserting the vertices one by one and connecting each vertex to the closest previously-inserted vertex in each cone. We show that for any integer [Formula: see text], ordered [Formula: see text]-graphs with [Formula: see text] cones have a tight spanning ratio of [Formula: see text]. We also show that for any integer [Formula: see text], ordered [Formula: see text]-graphs with [Formula: see text] cones have a tight spanning ratio of [Formula: see text]. We provide lower bounds for ordered [Formula: see text]-graphs with [Formula: see text] and [Formula: see text] cones. For ordered [Formula: see text]-graphs with [Formula: see text] and [Formula: see text] cones these lower bounds are strictly greater than the worst case spanning ratios of their unordered counterparts. These are the first results showing that ordered [Formula: see text]-graphs have worse spanning ratios than unordered [Formula: see text]-graphs. Finally, we show that, unlike their unordered counterparts, the ordered [Formula: see text]-graphs with 4, 5, and 6 cones are not spanners.

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